I'm one who certainly believes the story, because I've seen it happen on more than one occasion.
Generally, patients with severe enough COPD to require oxygen are accustomed to a lower oxygen tension in their blood and walk around on a day-to-day basis with a SpO2 in the low to mid 80s.
On an inpatient, I will usually aim to keep a COPD patient's SpO2 between 88 and 92%. The nurses don't like that, though, so when one goes into the room to examine the patient, make sure the O2 hasn't been turned up too high.
Outpatients who are on O2 at home (whether continuous or intermittent or nightly) obviously do not get monitored. In the US, in order for medicare to pay for O2 at home, the patient must have a resting or exercise SpO2 < 88%, which must be demonstrated and documented. Again, usually the goal with oxygen is to keep the SpO2 above 88%.
Hope this helps.
jd